Radiative transfer deals with electromagnetic wave propagation and is often difficult to model because it requires elaborate calculations. The traditional technique for the formulation of complex coating (e.g., paint) mixtures for the purpose of color matching is Kubelka-Munk Theory. The Kubelka-Munk method is used to calculate a two-flux approximation for solving the complicated equations in radiative transfer theory. Such an approximation is oftentimes inadequate for formulating complex coating mixtures that contain metallic, pearlescent, and other special effect pigments.
The underlying idea for the two-flux approximation is to find the diffuse radiance while solving the full radiative transfer equation. The approximation comes in with the approach to the full equation, however, because radiation fluxes are treated as angular-averaged properties, so one assumes that the details of the variation of the intensity are not very important for the predictions of these quantities, i.e. the parameters of color do not travel with viewing angle.
Many formulation strategies operate by working through every combination of, for example, four tinters out of ten, and determining the best match possible with each combination, and then looking for the best of the group. Other formulation strategies rely on neural networks, which reduce computation time, but are still fundamentally brute-force strategies.
Thus, a need exists for systems and methods that are suitable for analyzing complex coating mixtures containing effect pigments, for example metallic and pearlescent pigments.